2014
DOI: 10.1007/s00205-014-0799-9
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Weak Solutions to Fokker–Planck Equations and Mean Field Games

Abstract: We deal with systems of PDEs, arising in mean field games theory, where viscous Hamilton–Jacobi and Fokker–Planck equations are coupled in a forward- backward structure. We consider the case of local coupling, when the running cost depends on the pointwise value of the distribution density of the agents, in which case the smoothness of solutions is mostly unknown. We develop a complete weak theory, proving that those systems are well-posed in the class of weak solutions for monotone couplings under general gro… Show more

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Cited by 138 publications
(151 citation statements)
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“…However, thanks to the representation formula (26), it admits an extension operator that is well defined on L 2 (0, T ; L 2 (Ω; R d )). With this, Proposition 12, and the variational inequality dJ(ū)(u −ū) ≥ 0, which holds for any u ∈ U ad and locally optimal solutionū, we deduce the system of first order necessary optimality conditions.…”
Section: Proposition 12mentioning
confidence: 99%
See 2 more Smart Citations
“…However, thanks to the representation formula (26), it admits an extension operator that is well defined on L 2 (0, T ; L 2 (Ω; R d )). With this, Proposition 12, and the variational inequality dJ(ū)(u −ū) ≥ 0, which holds for any u ∈ U ad and locally optimal solutionū, we deduce the system of first order necessary optimality conditions.…”
Section: Proposition 12mentioning
confidence: 99%
“…In recent years, the well-posedness of the FP equation under low regularity assumptions on the coefficients has been studied in connection with existence, uniqueness and stability of martingale solutions to the related stochastic differential equation [22,11]. Furthermore, control properties of the FP equation have become of main interest in mean field game theory; see [26]. Our focus is on the optimal control of the FP equation.…”
Section: Introductionmentioning
confidence: 99%
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“…We use Hilbert space techniques (similarly to the one developed in [9,3,34]; see also Section 3.1. from [35]) to study the uniqueness of a solution of the diffusive crowd motion model with density constraints described in Subsection 1.2 (see also [31]). Moreover we can expect that this holds under more general assumptions in the presence of a non-degenerate diffusion in the model.…”
Section: Bounded Vector Field In the Diffusive Casementioning
confidence: 99%
“…As far as we are aware of, this question is a missing puzzle in full generality in the models studied in [25,26,27,31]. Let us remark that the uniqueness question is crucial if one wants to include this type of models into a larger system and one aims to show existence results by fixed point methods, as it is done for Mean Field Games in general for instance (for example as in [35]). …”
Section: Introductionmentioning
confidence: 99%